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kwisangqaquqw3

kwisangqaquqw3

Answered question

2022-05-15

How to simplify ( cos x sin x ) ( 2 tan x + 1 cos x ) + 2 = 0 to get a general solution?

Answer & Explanation

pulpasqsltl

pulpasqsltl

Beginner2022-05-16Added 18 answers

Verify that x = π ( 2 m + 1 ) , m Z , , are not solutions to the equation, then use the tangent half-angle substitution t = tan x 2 to rewrite the equation in terms of t (for tanx use the double-angle identity):
( 1 t 2 1 + t 2 2 t 1 + t 2 ) ( 2 2 t 1 t 2 + 1 + t 2 1 t 2 ) + 2 = 0
After clearing denominators and factorizing, this simplifies to
( 3 t 2 1 ) ( t 2 + 2 t + 3 ) = 0
So the solutions are t = tan x 2 = ± 1 3 which gives us
x 2 = ± π 6 + k π x = ± π 3 + 2 k π , k Z

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